Stable temp is important beyond just staying in the range. Yeast metabolism is largely, or almost entirely, controlled by temperature. The hotter the yeast get and the more active they become, but at the higher activity levels they also produce additional esters (peachy flavors) and phenolics (banana and clove flavors). So the same yeast in the same wort with a recommended temp range of 60-72, one fermented at a consistent 60 and the other at a consistent 70, will be significantly different beers. That temp range is only a guide and generally, unless you want to encourage ester or phenolic production, you'll get a cleaner tasting beer at the lower end of that range.
But beyond even that, what I have noticed is that the direction the temp is going has an impact. For instance, if I get my wort fermenting at 70 (same 60-72 temp range) and then drop the temp rapidly to 64 the yeast will become much less active. Something about a rapidly decreasing temp tells them to get ready for nap time. They begin to fall out of suspension. Then the temp goes back up, but the beer has some alcohol in it now and less food, so it's harder for them to get active. Alcohol is yeast waste product and they have a hard time in a high alcohol environment as far as fermentation is concerned. So you're likely to cost yourself a point or two on your F.G. by starting high, dropping rapidly and then going high again. Once the yeast become active you really want them to keep on chuggin until they are done. In the same situation but starting at 62 and waiting about three days until the yeast activity just starts to slow down, you can now ramp the temp up to keep them going a little longer and maybe get an extra point off your F.G..
So to go up and down kind of sends mixed signals to the yeast and your results will be less predictable.
And in my opinion, the temp ranges are always too wide. I frequently ferment BELOW the recommended temp range for my ales, but only by a couple degrees. Now I make really big starters and oxygenate with pure O2, so the yeast are in a good position to get the job done even with the low temp sort of suppressing their activity. It depends on what you're trying to do. But my basic rule of thumb is that the closer to the bottom of the recommended range the better if you want a clean tasting beer.